In order to represent, compute and reason with advanced data types one must go beyond the traditional treatment of data types as being inductive types and, instead, consider them as inductive families. Strictly positive types (SPTs) form a grammar for defining inductive types and, consequently, a fundamental question in the the theory of inductive families is what constitutes a corresponding grammar for inductive families.
This paper answers this question in the form of strictly positive families or SPFs. We show that these SPFs can be used to represent and compute with a variety of advanced data types, that generic programs can naturally be written over the universe of SPFs and that SPFs have a normal form in terms of indexed containers which are based upon the shapes and positions metaphor. Finally, we validate ou computational perspective by implementing SPFs in the programming language Epigram and, further, comment on how SPFs provide a meta-language for Epigram's data types.
为了用高级数据类型表示,计算和推理,必须超越传统的将数据类型视为归纳类型的处理,而是将其视为归纳族。严格肯定类型(SPT)构成了定义归纳类型的语法,因此,归纳族理论中的一个基本问题是什么构成了归纳族的相应语法。 P>
本文以严格阳性家庭 I>或SPF的形式回答了这个问题。我们证明了这些SPF可以用于表示和计算各种高级数据类型,可以自然地在SPF的整个域上编写通用程序,并且SPF在基于形状的索引容器方面具有正常形式和职位隐喻。最后,我们通过在编程语言Epigram中实现SPF来验证计算的观点,并进一步评论SPF如何为Epigram的数据类型提供元语言。 P>
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